MATH32051
HYPERBOLIC GEOMETRY
January 2022
1.
(i) Let
Prove that Γ is a subgroup of M¨ob(H).
(You may use facts from the course about composing M¨obius transformations and matrices without proof, provided that you state them clearly.) [10 marks]
2.
(i) State, without proof, the Gauss-Bonnet Theorem for a hyperbolic polygon. [2 marks]
(ii) Consider the diagram below. Let e ≥ 1. Here, the semi-circle between ie and 1 is a geodesic. By drawing in the (Euclidean) centre and (Euclidean) radius of the semi-circle passing through 1 and i` and considering a (Euclidean) right-angled triangle, show that [6 marks]
(iii) Consider the hyperbolic quadrilateral P with vertices at −1, 0, 1, i√
3 below. Determine (in the form. aπ/b for suitable integers a, b) the hyperbolic area of P, giving reasons for your answer.
(You may use the fact that sin π/3 = √
3/2 without proof.) [6 marks]
3.
(i) Let γ1, γ2 ∈ M¨ob(H). What does it mean to say that γ1, γ2 are conjugate? [2 marks]
(ii) Recall that a M¨obius transformation γ ∈ M¨ob(H) is said to be parabolic if γ has a unique fixed point on ∂H and no fixed points in H. The following result was proved in the course:
Let γ ∈ M¨ob(H) and suppose that γ = id. Then the following are equivalent:
(1) γ is parabolic
(2) τ (γ) = 4
(3) γ is conjugate to a translation z 7→ z + b
(4) γ is conjugate either to z 7→ z + 1 or to z 7→ z − 1.
Prove that (1) ⇒ (3). (You do not need to prove the other implications.) [8 marks]
(iii) Suppose that γ ∈ M¨ob(H) is parabolic. Is γ
−1 parabolic? Give either a proof or a counter-example. [4 marks]
(iv) Letγ1(z) = z + β, β ∈ R, β = 0, be a translation. Suppose that γ2 ∈ M¨ob(H) commutes with γ1 (so that γ1γ2(z) = γ2 γ1 (z) for all z). Show that if γ2 commutes with γ1 then γ2 is also a translation. [6 marks]
4.
(i) Let Γ be a Fuchsian group acting on H. What does it mean to say that a set F ⊂ H is a fundamental domain for Γ? [3 marks]
(ii) Briefly explain how to construct a Dirichlet region for a Fuchsian group Γ. (Your answer should include a statement of how to choose p, a definition of the terms [p, γ(p)], Lp(γ), Hp(γ) and D(p) that were defined in the course.) [5 marks]
(iii) Let z1 = x1 + iy1, z2 = x2 + iy2 ∈ H. It was proved in the course that the perpendicular bisector of [z1, z2], the arc of geodesic from z1 to z2, is given by {z ∈ H | dH(z, z1) = dH(z, z2)}. It was also shown in the course that this can be rewritten as
(You do not need to prove this.)
Use (∗) to show that the perpendicular bisector of [1 + i, 4
n + 4n
i] is
[6 marks]
(iv) Let Γ = {γn | γn(z) = 4n
z, n ∈ Z}. Let p = 1 + i. Use the result of (iii) to calculate D(p). [4 marks]
(v) Let Γ be as in (iv). Is there a fundamental domain F for Γ with AreaH(F) < ∞? Briefly justify your answer. [4 marks]
5.
(i) Let E = v0 → v1 → · · · → vn−1 be an elliptic cycle with corresponding elliptic cycle transformation γv0,s0. What does it mean to say that E satisfies the Elliptic Cycle Condition?
Let P = v0 → v1 → · · · → vn−1 be a parabolic cycle with corresponding parabolic cycle transformation γv0,s0. What does it mean to say that P satisfies the Parabolic Cycle Condition? [4 marks]
(ii) Consider the polygon P illustrated below with sides paired as indicated. (Here all 3 lines are geodesics. An extra vertex has been introduced at i/4 so that no side is paired with itself.)
Let k > 0. Let
Verify that γ1, γ2 pair the sides as indicated. [4 marks]
(iii) Determine the elliptic cycles and parabolic cycles. For each elliptic cycle, determine the elliptic cycle transformation and the angle sum along the elliptic cycle. For each parabolic cycle, determine the parabolic cycle transformation. [10 marks]
(iv) Determine the value of k for which γ1, γ2 generate a Fuchsian group Γ? In this case, give a presentation of Γ in terms of generators and relations. [10 marks]
(vi) For the Fuchsian group Γ determined in (iv), how many distinct words in Γ are there are of length 3? Briefly explain your reasoning. [6 marks]